Hostname: page-component-848d4c4894-pftt2 Total loading time: 0 Render date: 2024-05-31T13:18:44.229Z Has data issue: false hasContentIssue false
  • English
  • Français

Acoustic resonances and trapped modes in annular plate cascades

Published online by Cambridge University Press:  01 June 2009

WERNER KOCH
WERNER KOCH*
Affiliation:
Institute of Aerodynamics and Flow Technology, DLR Göttingen 37073, Germany
*
†Email address for correspondence: werner.koch@dlr.de
Get access Rights & Permissions [Opens in a new window]

Abstract

As a stepping stone towards understanding acoustic resonances in axial flow compressors, acoustic resonances are computed numerically in fixed single and tandem plate cascades in an infinitely long annular duct. Applying perfectly matched layer absorbing boundary conditions in the form of the complex scaling method of atomic and molecular physics to approximate the radiation condition the resonance problem is transformed into an eigenvalue problem. Of particular interest are resonances with zero radiation damping (trapped modes) or very small radiation damping (nearly trapped modes). Such resonances can be excited by wakes from compressor cascades or struts. If the shedding frequency is sufficiently close to an acoustic resonant frequency, the latter may control the vortex shedding causing high-intensity tonal noise or occasionally even blade failure. All resonances are computed for zero mean flow approximating low-Mach-number flows. The influence of various cascade parameters on the resonant frequencies is studied and, whenever possible, our numerical results are compared with published experimental findings.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 628 , 10 June 2009 , pp. 155 - 180
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abramowitz, M. & Stegun, I. 1965 Handbook of Mathematical Functions. Dover.Google Scholar
Aguilar, J. & Combes, J. 1971 A class of analytic perturbations for one-body Schrödinger Hamiltonians. Commun. Math. Phys. 22, 269–279.CrossRefGoogle Scholar
Baslev, E. & Combes, J. 1971 Spectral properties of many body Schrödinger operators with dilation analytic interactions. Commun. Math. Phys. 22, 280–294.CrossRefGoogle Scholar
Béerenger, J. 1994 A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114, 185–200.CrossRefGoogle Scholar
Berry, M. & Wilkinson, M. 1984 Diabolical points in the spectra of triangles. Proc. R. Soc. Lond. A 392 (1802), 15–43.Google Scholar
Callan, M., Linton, C. & Evans, D. 1991 Trapped modes in two-dimensional waveguides. J. Fluid Mech. 229, 51–64.CrossRefGoogle Scholar
Camp, T. 1999 A study of acoustic resonance in a low-speed multistage compressor. ASME J. Turbomach. 121, 36–43.CrossRefGoogle Scholar
Chew, W. & Weedon, W. 1994 A 3-D perfectly matched medium from modified Maxwell's equation with stretched coordinates. Microw. Opt. Technol. Lett. 7 (13), 599–604.CrossRefGoogle Scholar
Collino, F. & Monk, P. 1998 The perfectly matched layer in curvilinear coordinates. SIAM J. Sci. Comput. 19 (6), 2061–2090.CrossRefGoogle Scholar
Cooper, A. & Peake, N. 2006 Rotor-stator interaction noise in swirling flow: stator sweep and lean effects. AIAA J. 44 (5), 981–991.CrossRefGoogle Scholar
Cumpsty, N. & Whitehead, D. 1971 The excitation of acoustic resonances by vortex shedding. J. Sound Vib. 18, 353–369.CrossRefGoogle Scholar
Duan, Y. 2004 Trapped modes and acoustic resonances. PhD thesis, Loughborough University, Loughborough.Google Scholar
Duan, Y., Koch, W., Linton, C. & McIver, M. 2007 Complex resonances and trapped modes in ducted domains. J. Fluid Mech. 571, 119–147.CrossRefGoogle Scholar
Duan, Y. & McIver, M. 2004 Rotational acoustic resonances in cylindrical waveguides. Wave Motion 39, 261–274.CrossRefGoogle Scholar
Duclos, P. & Exner, P. 1995 Curvature-induced bound states in quantum waveguides in two and three dimensions. Rev. Math. Phys. 7 (1), 73–102.CrossRefGoogle Scholar
Elhadidi, B. & Atassi, H. 2005 Passive control for turbofan tonal noise. AIAA J. 43 (11), 2279–2292.CrossRefGoogle Scholar
Envia, E. & Nallasamy, M. 1999 Design selection and analysis of a swept and leaned stator concept. J. Sound Vib. 228 (4), 793–836.CrossRefGoogle Scholar
Evans, D. 1992 Trapped acoustic modes. IMA J. Appl. Math. 49 (1), 45–60.CrossRefGoogle Scholar
Evans, D., Levitin, M. & Vassiliev, D. 1994 Existence theorems for trapped modes. J. Fluid Mech. 261, 21–31.CrossRefGoogle Scholar
Evans, D. & Linton, C. 1991 Trapped modes in open channels. J. Fluid Mech. 225, 153–175.CrossRefGoogle Scholar
Evans, D. & Linton, C. 1994 Acoustic resonance in ducts. J. Sound Vib. 173, 85–94.CrossRefGoogle Scholar
Fric, T., Villarreal, R., Auer, R., James, M., Ozgur, D. & Staley, T. 1998 Vortex shedding from struts in annular exhaust diffuser. ASME J. Turbomach. 120, 186–192.CrossRefGoogle Scholar
Harari, I., Patlashenko, I. & Givoli, D. 1998 Dirichlet-to-Neumann maps for unbounded wave guides. J. Comput. Phys. 143, 200–223.CrossRefGoogle Scholar
Hein, S., Hohage, T. & Koch, W. 2004 On resonances in open systems. J. Fluid Mech. 506, 255–284.CrossRefGoogle Scholar
Hein, S., Hohage, T., Koch, W. & Schöberl, J. 2007 Acoustic resonances in a high lift configuration. J. Fluid Mech. 582, 179–202.CrossRefGoogle Scholar
Hein, S. & Koch, W. 2008 Acoustic resonances and trapped modes in pipes and tunnels. J. Fluid Mech. 605, 401–428.CrossRefGoogle Scholar
Hellmich, B. 2008 Acoustic resonance in a high-speed axial compressor. PhD thesis, Universität Hannover, Hannover.CrossRefGoogle Scholar
Hellmich, B. & Seume, J. 2008 Causes of acoustic resonances in a high-speed axial compressor. ASME J. Turbomach. 130, 031003-1–031003-9.CrossRefGoogle Scholar
Hislop, P. & Sigal, I. 1996 Introduction to Spectral Theory. Springer.CrossRefGoogle Scholar
Hu, F. 2004 Absorbing boundary conditions. Intl J. Comput. Fluid Dyn. 18 (6), 513–522.CrossRefGoogle Scholar
Johnson, C. & Loehrke, R. 1984 An experimental investigation of wake edge tones. AIAA J. 22, 1249–1253.CrossRefGoogle Scholar
Kim, S. & Pasciak, J. 2009 The computation of resonances in open systems using a perfectly matched layer. Math. Comp. S0025–5718(09)02227-3.Google Scholar
Koch, W. 1983 Resonant acoustic frequencies of flat plate cascades. J. Sound Vib. 88, 233–242.CrossRefGoogle Scholar
Koch, W. 2005 Acoustic resonances in rectangular open cavities. AIAA J. 43 (11), 2342–2349.CrossRefGoogle Scholar
Lamoureux, P. & Weaver, D. 1991 The effects of turbulence and damping on pipeline acoustic resonance. Proc. Inst. Mech. Engrs., Part C 205, 303–312.Google Scholar
Legerton, M. 1992 Flow induced acoustic resonances. PhD thesis, University of Wales, Swansea.Google Scholar
Legerton, M., Stoneman, S. & Parker, R. 1991 An experimental investigation into flow induced acoustic resonances in an annular cascade. Proc. Inst. Mech. Engrs., Part C 205, 445–452.Google Scholar
Li, Y. & Mei, C. 2006 Subharmonic resonance of a trapped wave near a vertical cylinder in a channel. J. Fluid Mech. 561, 391–416.CrossRefGoogle Scholar
Linton, C. & McIver, P. 1998 Acoustic resonances in the presence of radial fins in circular cylindrical waveguides. Wave Motion 28, 99–117.CrossRefGoogle Scholar
Linton, C. & McIver, M. 2002 Periodic structures in waveguides. Proc. R. Soc. Lond. A 458, 3003–3021.CrossRefGoogle Scholar
Linton, C. & Ratcliffe, K. 2004 Bound states in coupled guides. I. Two dimensions. J. Math. Phys. 45 (4), 1359–1379.CrossRefGoogle Scholar
Maniar, H. & Newman, J. 1997 Wave diffraction by a long array of cylinders. J. Fluid Mech. 339, 309–330.CrossRefGoogle Scholar
Mohany, A. & Ziada, S. 2005 Flow-excited acoustic resonance of two tandem cylinders in cross-flow. J. Fluids Struct. 21 (1), 103–119.CrossRefGoogle Scholar
Moiseyev, N. 1998 Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling. Phys. Rep. 302, 211–293.CrossRefGoogle Scholar
Nayfeh, A. & Huddleston, D. 1979 Resonant acoustic frequencies of parallel plates (AIAA–Paper 79–1522). In Twelfth AIAA Fluid and Plasma Dynamics Conference, Williamsburg, VA.Google Scholar
Parker, R. 1966 Resonance effects in wake shedding from parallel plates: some experimental observations. J. Sound Vib. 4 (1), 62–72.CrossRefGoogle Scholar
Parker, R. 1967 a Resonance effects in wake shedding from compressor blading. J. Sound Vib. 6, 302–309.CrossRefGoogle Scholar
Parker, R. 1967 b Resonance effects in wake shedding from parallel plates: calculation of resonant frequencies. J. Sound Vib. 5, 330–343.CrossRefGoogle Scholar
Parker, R. 1968 An investigation of acoustic resonance effects in an axial flow compressor stage. J. Sound Vib. 8, 281–297.CrossRefGoogle Scholar
Parker, R. 1984 Acoustic resonances and blade vibration in axial flow compressors. J. Sound Vib. 92, 529–539.CrossRefGoogle Scholar
Parker, R. 1997 Aeroacoustics. Intl J. Fluid Dyn. 1, (http://elecpress.monash.edu.au/IJFD), Article 1.Google Scholar
Parker, R. & Griffiths, W. 1968 Low frequency resonance effects in wake shedding from parallel plates. J. Sound Vib. 7, 371–379.CrossRefGoogle Scholar
Parker, R. & Pryce, D. 1974 Wake excited resonances in an annular cascade: an experimental investigation. J. Sound Vib. 37, 247–261.CrossRefGoogle Scholar
Parker, R. & Stoneman, S. 1985 An experimental investigation of the generation and consequences of acoustic waves in an axial-flow compressor: large axial spacing between blade rows. J. Sound Vib. 99 (2), 169–182.CrossRefGoogle Scholar
Parker, R. & Stoneman, S. 1987 An experimental investigation of the generation and consequences of acoustic waves in an axial-flow compressor: the effect of variations in the axial spacing between blade rows. J. Sound Vib. 116 (3), 509–525.CrossRefGoogle Scholar
Parker, R. & Stoneman, S. 1989 The excitation and consequences of acoustic resonances in enclosed fluid flow around solid bodies. Proc. Inst. Mech. Engrs 203, 9–19.CrossRefGoogle Scholar
Porter, R. 2007 Trapped modes in thin elastic plates. Wave Motion 45, 3–15.CrossRefGoogle Scholar
Porter, R. & Evans, D. 1999 Rayleigh–Bloch surface waves along periodic gratings and their connection with trapped modes in waveguides. J. Fluid Mech. 386, 233–258.CrossRefGoogle Scholar
Ragab, S. & Salem-Said, A.-H. 2007 Response of a flat-plate cascade to incident vortical waves. AIAA J. 45 (9), 2140–2148.CrossRefGoogle Scholar
Rockwell, D. 1983 Oscillations of impinging shear layers. AIAA J. 21 (5), 645–664.CrossRefGoogle Scholar
Schöberl, J. 1997 NETGEN: an advancing front 2D/3D-mesh generator based on abstract rules. Comput. Vis. Sci. 1, 41–52.Google Scholar
Schulten, J. 1997 Vane sweep effects on rotor/stator interaction noise. AIAA J. 35 (6), 945–951.CrossRefGoogle Scholar
Simon, B. 1973 The theory of resonances for dilation analytic potentials and the foundations of time dependent perturbation theory. Ann. Math. 97, 247–274.CrossRefGoogle Scholar
Stoneman, S., Hourigan, K., Stokes, A. & Welsh, M. 1988 Resonant sound caused by flow past two plates in tandem in a duct. J. Fluid Mech. 192, 455–484.CrossRefGoogle Scholar
Tyler, J. & Sofrin, T. 1962 Axial compressor noise studies. Soc. Automot. Engrs Trans. 70, 309–332.Google Scholar
Ulbricht, I. 2002 Stabilität des stehenden Ringgitters. PhD thesis, Technische Universität Berlin, Berlin.Google Scholar
Ursell, F. 1951 Trapping modes in the theory of surface waves. Proc. Camb. Phil. Soc. 47, 347–358.CrossRefGoogle Scholar
Utsunomiya, T. & Eatock Taylor, R. 1999 Trapped modes around a row of circular cylinders in a channel. J. Fluid Mech. 386, 259–279.CrossRefGoogle Scholar
Woodley, B. & Peake, N. 1999 a Resonant acoustic frequencies of a tandem cascade. Part 1: zero relative motion. J. Fluid Mech. 393, 215–240.CrossRefGoogle Scholar
Woodley, B. & Peake, N. 1999 b Resonant acoustic frequencies of a tandem cascade. Part 2: rotating blade rows. J. Fluid Mech. 393, 241–256.CrossRefGoogle Scholar
Woodward, R., Elliott, D., Hughes, C. & Berton, J. 2001 Benefits of swept-and-leaned stators for fan noise reduction. J. Aircr. 38 (6), 1130–1138.CrossRefGoogle Scholar
Ziada, S., Oengören, A. & Vogel, A. 2002 Acoustic resonance in the inlet scroll of a turbo-compressor. J. Fluids Struct. 16 (3), 361–373.CrossRefGoogle Scholar